Problem: Integrate. $ \int \csc^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $-\cot(x)+C$ (Choice B) B $-\csc(x)+C$ (Choice C) C $-\sec(x)+C$ (Choice D) D $-\tan(x)+C$
Solution: We need a function whose derivative is $\csc^2(x)$. We know that the derivative of $\cot(x)$ is $-\csc^2(x)$, so let's start there: $\dfrac{d}{dx} \cot(x) = -\csc^2(x)$ Now let's multiply by $-1$ : $\dfrac{d}{dx} \left[ -1\cot(x) \right]= -1\dfrac{d}{dx} \cot(x) = \csc^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int \csc^2(x)\,dx =- \cot(x)\, + C$ The answer: $- \cot(x)\, + C$